Finite Group Databasebeta

← #163243

Subgroups — #163243

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20778C211yesmaximal, characteristic
4#7723C214yesmaximal, complemented
5#7720C214yesmaximal, complemented
6#7627C214yesmaximal
7C22×#125—21no—
8#2859—21no—
9#2859—21no—
10#2859—21no—
11#2859—21no—
12#2859C412yes—
13#2859C412yes—
14#2859C2212yes—
15#2859C2212yes—
16C2×#389—21no—
17C2×#389—21no—
18C2×#389C412yes—
19C2×#389C2212yes—
20C2×#384C411yescharacteristic
21C2×#384C411yescharacteristic
22C2×#382—21no—
23C2×C43C2211yescharacteristic
24#1839C2212yes—
25#1839C2212yes—
26#1831C2212yes—
27#1831C2212yes—
28#1817—21no—
29#1817—21no—
30#1797—21no—
31#1797—21no—
32#1767—21no—
33#1767—21no—
34#1749C2212yes—
35#1749C2212yes—
36#1745—21no—
37#1745—21no—
38#1271C22116yes—
39C2×#125—22no—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—48no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×#97—22no—
64C2×#97—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389C2×C414yes—
78#389C2×C414yes—
79#389C2×C414yes—
80#389C2314yes—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384C2×C412yes—
86#384C2×C412yes—
87#384C2×C412yes—
88#384C2×C412yes—
89#382—22no—
90#382—22no—
91#382—22no—
92#382—22no—
93C2×#95—22no—
94C2×#95—22no—
95C43C2×C411yescharacteristic
96C43C2×C411yescharacteristic
97C43D411yescharacteristic
98C43D411yescharacteristic
99C43D411yescharacteristic
100C43D411yescharacteristic
101C43C2311yescharacteristic
102C43C2311yescharacteristic
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336C2314yes—
110#336C2314yes—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C22×C8—24no—
121C22×C8—24no—
122#125—48no—
123#125—48no—
124#125—48no—
125#125—48no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—48no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—48no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2411yescharacteristic
185C4⋊C8—44no—
186C4⋊C8—44no—
187C22⋊C8—44no—
188C22⋊C8—44no—
189#97—44no—
190#97—44no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195C24C2×C22⋊C411yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4#10114yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C4⋊C4—28no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—48no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—48no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42AGL(1,ℤ/8)14yes—
288C42#13714yes—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C2×#10112yes—
294C23#42112yes—
295C23#42112yes—
296C23C22×C22⋊C411yescharacteristic
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—28no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—88no—
343C2×C4—88no—
344C2×C4#41012yes—
345C2×C4C2×#10112yes—
346C2×C4C2×#10112yes—
347C2×C4#41812yes—
348C2×C4#41812yes—
349C2×C4#41912yes—
350C2×C4#41912yes—
351C2×C4#56812yes—
352C2×C4C2×AGL(1,ℤ/8)12yes—
353C2×C4C2×#13712yes—
354C8—84no—
355C8—84no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22#153414yes—
363C22#287911yescharacteristic
364C22C2×#41011yescharacteristic
365C22C22×#10111yescharacteristic
366C22C2×#41811yescharacteristic
367C22C2×#41911yescharacteristic
368C22C2×#42111yescharacteristic
369C22C2×#42111yescharacteristic
370C4—22no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—84no—
385C4—84no—
386C2—24no—
387C2#1739411yescharacteristic
388C2#1749511yescharacteristic
389C2#1749611yescharacteristic
390C2#1758311yescharacteristic
391C2#1758411yescharacteristic
392C2C2×#153411yescharacteristic
393C2#1774211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7723C2semidirect
5. #7720C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#28592
#207782
#77232
#77202
#76272